Fibonacci and Related Sequences in Digital Filtering

In many communication and signal-processing systems, desired signals (sequences) are embedded in noise. Linear filters have been the primary tool for smoothing or recovering the desired signal from the degraded signal. Linear filters perform particularly well where the spectrum of the desired signal is significantly different from that of the interference. In many situations, however, the spectrum of the signal and of the noise are mixed in the same range and the performance of linear filters is very poor. Median filters can be used to circumvent these problems. Tukey [1] is generally credited with the idea of introducing nonlinear filters based on moving sample medians of the input signal. In this paper, we do not address the filtering problem, but we analyze the signal (sequence) set of median filtered binary sequences. To best explain the goal of this paper, the implementation of the median filter is described first. To begin, take a binary sequence of length n; across this signal we slide a window that spans 2s 1 samples of the binary sequence, for s = 2, 3, ... . At each point of the sequence, the median of the samples within the window of the filter is computed and the output of the filter at the center sample is set equal to the computed median. To account for start-up and end effects at the two endpoints of the n-length sequence, s 1 samples are appended to the beginning and end of the sequence. The value of the appended samples to the beginning is equal to the value of the first sample; similarly, the value of the appended samples to the end of the sequence equals the value of the last sample of the sequence. Figure 1(a) shows a binary signal of length 10 being filtered by a filter of window of size 3. The filtered signal is shown below. The appended samples are shown as crosses (X). Figure 1(b) shows the same sequence filtered by a filter of window size 5. Figure 1(c) shows similar results with a larger window. An interesting observation is that there exist sequences that are not modified by the median filter. Moreover, it has been shown that any finite input sequence, after repeated median filtering, will be reduced to one of these invariant sequences [2]. A sequence that is not modified by the filtering process is called a "root" sequence. The following theorem provides the upper bound on the number of successive filter passes necessary to reduce an input sequence to a root sequence [2]: